Relaxation Models and Finite Element Schemes for the Shallow Water Equations (original) (raw)
Related papers
1999
Numerical schemes for the partial differential equations used to characterize stiffly forced conservation laws are constructed and analyzed. Partial differential equations of this form are found in many physical applications including modeling gas dynamics, fluid flow, and combustion. Many difficulties arise when trying to approximate solutions to stiffly forced conservation laws numerically. Some of these numerical difficulties are investigated. A new class of numerical schemes is developed to overcome some of these problems. The numerical schemes are constructed using an infinite sequence of conservation laws. Restrictions are given on the schemes that guarantee they maintain a uniform bound and satisfy an entropy condition. For schemes meeting these criteria, a proof is given of convergence to the correct physical solution of the conservation law. Numerical examples are presented to illustrate the theoretical results.
A study of numerical methods for hyperbolic conservation laws with stiff source terms
Journal of Computational Physics, 1990
The proper modeling of nonequilibrium gas dynamics is required in certain regimes of hypersonic flow. For inviscid flow this gives a system of conservation laws coupled with source terms representing the chemistry. Often a wide range of time scales is present in the problem, leading to numerical difficulties as in stiff systems of ordinary differential equations. Stability can be achieved by using implicit methods, but other numerical difficulties are observed. The behavior of typical numerical methods on a model advection equation with a parameter-dependent source term is studied. Two approaches to incorporate the source terms are utilized: MacCormack type predictor-corrector methods with flux limiters and splitting methods in which the fluid dynamics and chemistry are handled in separate steps. Comparisons over a wide range of parameter values are made. On the whole, the splitting methods perform somewhat better. In the stiff case, a numerical phenomenon of incorrect propagation speeds of discontinuities is observed and explained. Similar behavior was reported by Colella, Majda, and Roytburd (SIAM J. Sci. Stat. Comput. 7, 1059 (1986)) on a model combustion problem. Using the model scalar equation, we show that this is due to the introduction of nonequilibrium values through numerical dissipation in the advection step.
A flux-limited second order scheme for hyperbolic conservation laws with source terms
The theoretical foundations of high-resolution TVD schemes for homogeneous scalar conservation laws and linear systems of conservation laws have been firmly established through the work of Harten [8], Sweby [13], and Roe [11]. These TVD schemes seek to prevent an increase in the total variation of the numerical solution, and are successfully implemented in the form of flux-limiters or slope limiters for scalar conservation laws and systems. However, their application to conservation laws with source terms is still not fully developed. In this work we analyze the properties of a second order, flux-limited version of the Lax-Wendroff scheme preserving steady states [5]. Our technique is based on a flux limiting procedure applied only to those terms related to the physical flow derivative.
Development of new and modified numerical methods for hyperbolic conservation laws
Numerical Heat Transfer, Part B: Fundamentals, 2019
The aim of the present work is to propose and modify several numerical methods including the classes of temporal discretization methods for hyperbolic conservation laws. The first order in space standard Lax approximation is updated to modified first-order and newly proposed third-order accurate approximation. Presently proposed methods can be coupled with the modified and newly proposed Lax approximations and this coupling make the methods conservative. Some additional new classes of explicit and implicit methods for PDEs in time are proposed. Additionally, some new methods are given to reduce oscillations in the solutions. These new methods of reducing oscillations provide the conditions for coupling of first and higher-order methods.
Efficient high-resolution relaxation schemes for hyperbolic systems of conservation laws
International Journal for Numerical Methods in Fluids, 2007
In this work we present an upwind based high resolution scheme using flux limiters. Based on the direction of flow we choose the smoothness parameter in such a way that it lead to a truly upwind scheme without loosing total variation diminishing (TVD) property for hyperbolic linear system where characteristic values can be of either sign. Here we present and justify the choice of smoothness parameter. The numerical flux function of high resolution scheme is constructed using wave speed splitting so that it results into a scheme which truly respects the physical hyperbolicity property.
An alternative procedure for approximating hyperbolic systems of conservation laws
Nonlinear Analysis: Real World Applications, 2008
This work presents an alternative numerical procedure for simulating a class of nonlinear hyperbolic systems, using Glimm's method for advancing in time. The standard procedure to implement this methodology suffers from the disadvantage of requiring a complete solution of the associated Riemann problem-a task, in general, not easily reached. The alternative procedure introduced in this article consists in approximating the solution of the associated Riemann problem by piecewise constant functions always satisfying the jump condition-thus circumventing the difficulty of solving the Riemann problem and giving rise to an approximation easier to implement with lower computational cost. In order to illustrate the good performance of the alternative methodology proposed, two problems are considered-namely the transport of a pollutant in the atmosphere and the dynamics of the filling up of a rigid porous medium, modeled under a mixture theory viewpoint. Comparison with the standard procedure, employing the complete solution of the associated Riemann problem for implementing Glimm's scheme, has shown good agreement.
SIAM Journal on Numerical Analysis, 2004
We propose a class of finite element schemes for systems of hyperbolic conservation laws, that are based on finite element discretizations of appropriate relaxation models. We show that the finite element schemes are stable and, when the compensated compactness theory is applicable, do converge to a weak solution of the hyperbolic system. The schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. We also prove that the rate of convergenece of the relaxation system to a smooth solution of the conservation laws is of order O(ε).
The shallow water equations: An example of hyperbolic system
2008
Many problems of river management and civil protection consist of the evaluation of the maximum water levels and discharges that may be attained at particular locations during the development of an exceptional meteorological event. There is also the prevision of the scenario subsequent to the almost instantaneous release of a great volume of liquid. The situation is that of the breaking of a man made dam. There is therefore a necessity to develop adequate numerical models able to reproduce situations originated by the irregularities of a non-prismatic bed. It is also necessary to trace their applicability considering the difficulty of developing a model capable of producing solutions of the complete equations despite the irregular character of the river bed. When trying to use mathematical models as a predictive tool in the simulation of free surface flows, the hypothesis of one-dimensional models are not always valid. Such is the case when dealing with compound, or highly irregular, cross-section configurations, abrupt contractions and expansions, or rivers of high curvature. When trying to reproduce these hydraulic situations, it becomes necessary to use a two-dimensional formalism which takes into consideration the influence of transverse components of the flow. Many efforts have been recently devoted to the development of multidimensional techniques for free surface flows.
Computers & Mathematics with Applications, 2000
We propose a way to construct robust numerical schemes for the computations of numerical solutions of one- and two-dimensional hyperbolic systems of balance laws. In order to reduce the computational cost, we selected the family of flux vector splitting schemes. We reformulate the source terms as nonconservative products and treat them directly in the definition of the numerical fluxes by means of generalized jump relations. This is applied to a 1D shallow water system with topography and to a 2D simplified model of two-phase flows with damping effects. Numerical results and comparisons with a classical centered discretizations scheme are supplied.